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Questions and Answers
What is a linear combination of vectors?
What is a spanning set of a vector space?
A set where every vector in the space can be represented as a linear combination of its vectors.
Define the span of a set.
The set of all linear combinations of the vectors in the set.
Span(S) is a subspace of V.
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When is a set of vectors called linearly independent?
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A set S is called linearly dependent if it has _____ solutions.
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What are the steps to test for linear independence?
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A set is linearly independent if one of the vectors is a linear combination of others.
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Two vectors in a vector space are linearly independent if one is a scalar multiple of the other.
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Study Notes
Linear Combinations
- A vector v in a vector space V is a linear combination of vectors u1, u2, ..., uk if it can be expressed as c1u1 + c2u2 + ... + ckuk, where c1, c2, ..., ck are scalars.
Spanning Sets
- A subset S = {v1, v2, ..., vk} of a vector space V is a spanning set if every vector in V can be expressed as a linear combination of vectors in S.
- If S spans V, it means that the entirety of V can be reached by taking linear combinations of vectors in S.
Span of a Set
- The span of a set of vectors S = {v1, v2, ..., vk} is defined as the collection of all linear combinations of the vectors in S.
- Denoted as span(S) or span{v1, v2, ..., vk}, it represents all possible linear combinations using real numbers as coefficients.
- If span(S) = V, then it is stated that V is spanned by S.
Subspace Properties
- Span(S) is a subspace of the vector space V that includes all linear combinations of vectors in S.
- It is the smallest subspace containing S, meaning any other subspace that contains S must also include span(S).
Linear Dependence and Independence
- A set of vectors S = {v1, v2, ..., vk} is linearly independent if the equation c1v1 + c2v2 + ... + ckvk = 0 only holds for the trivial solution c1 = 0, c2 = 0, ..., ck = 0.
- If there are nontrivial solutions (other than the trivial one), then the set S is linearly dependent.
Testing Dependence/Independence
- To test whether a set S is linearly independent or dependent:
- Set up the equation c1v1 + c2v2 + ... + ckvk = 0 and transform it into a system of linear equations.
- Apply Gaussian elimination to assess the uniqueness of the solution.
- If the solution is unique (only trivial), S is independent; if there are nontrivial solutions, it is dependent.
Theorem on Linear Independence
- A set of vectors S = {v1, v2, ..., vk}, with k â‰¥ 2, is linearly independent only if at least one vector can be expressed as a linear combination of the others.
Corollary on Two Vectors
- Two vectors u and v in a vector space V are linearly independent if and only if neither is a scalar multiple of the other.
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Test your knowledge on spanning sets and linear independence with these flashcards. Understand key concepts like linear combinations and spanning sets in vector spaces. Perfect for students studying linear algebra.